A new fractal dimension: The topological Hausdorff dimension

Abstract

Abstract. We introduce a new concept of dimension for metric spaces, the so-called topological Hausdorff dimension. It is defined by a very natural combination of the definitions of the topological dimension and the Hausdorff dimension. The value of the topological Hausdorff dimension is always between the topological dimension and the Hausdorff dimension, in particular, this new dimension is a non-trivial lower estimate for the Hausdorff dimension.We examine the basic properties of this new notion of dimension, compare it to other well-known notions, determine its value for some classical fractals such as the Sierpinski carpet, the von Koch snowflake curve, Kakeya sets, the trail of the Brownian motion, etc.As our first application, we generalize the celebrated result of Chayes, Chayes and Durrett about the phase transition of the connectedness of the limit set of Mandelbrot's fractal percolation process. They proved that certain curves show up in the limit set when passing a critical probability, and we prove that actually 'thick' families of curves show up, where roughly speaking the word thick means that the curves can be parametrized in a natural way by a set of large Hausdorff dimension. The proof of this is basically a lower estimate of the topological Hausdorff dimension of the limit set. For the sake of completeness, we also give an upper estimate and conclude that in the non-trivial cases the topological Hausdorff dimension is almost surely strictly below the Hausdorff dimension.Finally, as our second application, we show that the topological Hausdorff dimension is precisely the right notion to describe the Hausdorff dimension of the level sets of the generic continuous function (in the sense of Baire category) defined on a compact metric space.

Inductive topological Hausdorff dimensions and fibers of generic continuous functions

“…This implies that these dimensions are new and independent in the following sense: The nth inductive topological Hausdorff dimension is not the function of the topological dimension and the kth inductive topological Hausdorff dimensions, where k runs over N \ {n}. This generalizes a theorem in [1] concerning the possible values of the topological Hausdorff dimension. The material developed here will be not used in the subsequent sections.…”

“…In Section 5 we completely describe the possible values of the inductive topological Hausdorff dimensions based on ideas from [1]. This is a supplement for the theory of the dimensions, we will not use this result in the subsequent sections.…”

“…For more information on topological Hausdorff dimension see [1], here we mention only the results concerning level sets of generic continuous functions.…”

“…The following theorem from [1] shows the connection between the topological Hausdorff dimension and the level sets of the generic f ∈ C 1 (K). Theorem 1.4 (Balka, Buczolich, Elekes).…”

“…The topological Hausdorff dimension (introduced in [1]) is defined analogously to the topological dimension. However, it is not inductive, and it can attain non-integer values as well.…”